Degeneracy and Colorings of Squares of Planar Graphs without 4-Cycles

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(\mathrm{\Delta }(G)+72)$-degenerate. This implies an upper bound of $\mathrm{\Delta }(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon–Tarsi number, and correspondence chromatic number. We also show that if $\mathrm{\Delta }(G)$ is sufficiently large, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $\mathrm{\Delta }(G)+2$ (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let $S$ be a finite list of positive integers, with $4\mathrm{\notin }S$. For each constant $C$, we construct a planar graph $G_{S,C}$ with no cycle with length in $S$, but for which $\chi (G_{S,C}^2)>\mathrm{\Delta }(G_{S,C})+C$.